Probability density function
Cumulative distribution function
Parameters (real)
Mean infinite
Variance infinite
Skewness does not exist
Ex. kurtosis does not exist
MGF does not exist

In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution.[1]


Probability density function

The log-Cauchy distribution has the probability density function:

where is a real number and .[1][2] If is known, the scale parameter is .[1] and correspond to the location parameter and scale parameter of the associated Cauchy distribution.[1][3] Some authors define and as the location and scale parameters, respectively, of the log-Cauchy distribution.[3]

For and , corresponding to a standard Cauchy distribution, the probability density function reduces to:[4]

Cumulative distribution function

The cumulative distribution function (cdf) when and is:[4]

Survival function

The survival function when and is:[4]

Hazard rate

The hazard rate when and is:[4]

The hazard rate decreases at the beginning and at the end of the distribution, but there may be an interval over which the hazard rate increases.[4]


The log-Cauchy distribution is an example of a heavy-tailed distribution.[5] Some authors regard it as a "super-heavy tailed" distribution, because it has a heavier tail than a Pareto distribution-type heavy tail, i.e., it has a logarithmically decaying tail.[5][6] As with the Cauchy distribution, none of the non-trivial moments of the log-Cauchy distribution are finite.[4] The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation.[7][8]

The log-Cauchy distribution is infinitely divisible for some parameters but not for others.[9] Like the lognormal distribution, log-t or log-Student distribution and Weibull distribution, the log-Cauchy distribution is a special case of the generalized beta distribution of the second kind.[10][11] The log-Cauchy is actually a special case of the log-t distribution, similar to the Cauchy distribution being a special case of the Student's t distribution with 1 degree of freedom.[12][13]

Since the Cauchy distribution is a stable distribution, the log-Cauchy distribution is a logstable distribution.[14] Logstable distributions have poles at x=0.[13]

Estimating parameters

The median of the natural logarithms of a sample is a robust estimator of .[1] The median absolute deviation of the natural logarithms of a sample is a robust estimator of .[1]


In Bayesian statistics, the log-Cauchy distribution can be used to approximate the improper Jeffreys-Haldane density, 1/k, which is sometimes suggested as the prior distribution for k where k is a positive parameter being estimated.[15][16] The log-Cauchy distribution can be used to model certain survival processes where significant outliers or extreme results may occur.[2][3][17] An example of a process where a log-Cauchy distribution may be an appropriate model is the time between someone becoming infected with HIV and showing symptoms of the disease, which may be very long for some people.[3] It has also been proposed as a model for species abundance patterns.[18]


  1. ^ a b c d e f Olive, D.J. (June 23, 2008). "Applied Robust Statistics" (PDF). Southern Illinois University. p. 86. Archived from the original (PDF) on September 28, 2011. Retrieved 2011-10-18.
  2. ^ a b Lindsey, J.K. (2004). Statistical analysis of stochastic processes in time. Cambridge University Press. pp. 33, 50, 56, 62, 145. ISBN 978-0-521-83741-5.
  3. ^ a b c d Mode, C.J. & Sleeman, C.K. (2000). Stochastic processes in epidemiology: HIV/AIDS, other infectious diseases. World Scientific. pp. 29–37. ISBN 978-981-02-4097-4.
  4. ^ a b c d e f Marshall, A.W. & Olkin, I. (2007). Life distributions: structure of nonparametric, semiparametric, and parametric families. Springer. pp. 443–444. ISBN 978-0-387-20333-1.
  5. ^ a b Falk, M.; Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer. p. 80. ISBN 978-3-0348-0008-2.
  6. ^ Alves, M.I.F.; de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF). Archived from the original (PDF) on June 23, 2007.
  7. ^ "Moment". Mathworld. Retrieved 2011-10-19.
  8. ^ Wang, Y. "Trade, Human Capital and Technology Spillovers: An Industry Level Analysis". Carleton University: 14. Cite journal requires |journal= (help)
  9. ^ Bondesson, L. (2003). "On the Lévy Measure of the Lognormal and LogCauchy Distributions". Methodology and Computing in Applied Probability: 243–256. Archived from the original on 2012-04-25. Retrieved 2011-10-18.
  10. ^ Knight, J. & Satchell, S. (2001). Return distributions in finance. Butterworth-Heinemann. p. 153. ISBN 978-0-7506-4751-9.
  11. ^ Kemp, M. (2009). Market consistency: model calibration in imperfect markets. Wiley. ISBN 978-0-470-77088-7.
  12. ^ MacDonald, J.B. (1981). "Measuring Income Inequality". In Taillie, C.; Patil, G.P.; Baldessari, B. (eds.). Statistical distributions in scientific work: proceedings of the NATO Advanced Study Institute. Springer. p. 169. ISBN 978-90-277-1334-6.
  13. ^ a b Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Science. Wiley. pp. 101–102, 110. ISBN 978-0-471-15064-0.
  14. ^ Panton, D.B. (May 1993). "Distribution function values for logstable distributions". Computers & Mathematics with Applications. 25 (9): 17–24. doi:10.1016/0898-1221(93)90128-I.
  15. ^ Good, I.J. (1983). Good thinking: the foundations of probability and its applications. University of Minnesota Press. p. 102. ISBN 978-0-8166-1142-3.
  16. ^ Chen, M. (2010). Frontiers of Statistical Decision Making and Bayesian Analysis. Springer. p. 12. ISBN 978-1-4419-6943-9.
  17. ^ Lindsey, J.K.; Jones, B. & Jarvis, P. (September 2001). "Some statistical issues in modelling pharmacokinetic data". Statistics in Medicine. 20 (17–18): 2775–278. doi:10.1002/sim.742. PMID 11523082.
  18. ^ Zuo-Yun, Y.; et al. (June 2005). "LogCauchy, log-sech and lognormal distributions of species abundances in forest communities". Ecological Modelling. 184 (2–4): 329–340. doi:10.1016/j.ecolmodel.2004.10.011.